Polynomial equation solver



Jan. 20, 1953 R. SERRELL Erm.

PoLYNoMrAL EQUATION som/ER Filed July 2l, 1949 MNM.

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l /M/E/vro/e HUBERT SERRELL x0 EDWIN A.EULDBERE ATTORNEY Patented Jan.20, 1953 UNITED STATES PATENT OFFICE POLYNOMIAL EQUATION SOLVER- WareApplication July 21, 1949, Serial No. 105,936

(Cl. 23S- 61) 2 Claims.

This invention relates to computing devices, and has for its principalpurpose the provision of an improved device and method of operationwhereby the real and complex roots of polynomials (algebraic equations)may be determined.

This improved device includes linear ampliers, linear potentiometers andnull indicators similar to those described in Brown and Goldberg PatentNo. 2,454,549- and Brown Patent No. 2,455,974. These amplifiers andpotentiometers are interconnected in such a manner as to build up (bydirect synthesis) the complex values of the polynomial whose roots aredesired. By means of two separate mechanical gangings of certain of thepotentiometers and two null indicators, the real and imaginary parts ofthe roots of the polynomial may be determined.

The invention will be better understood from the following descriptionconsidered in connection with the accompanying drawingsv and its scopeis indicated by the appended claims.

Referring to the drawings:

Fig. 1 is a circuit diagram illustrating a portion of the computerillustrated in Fig. 2, and included for the purpose of explaining howthe computer of Fig. 2 is utilizable for solving polynomials having realcoeflicients` and real roots, g

Fig. 2 illustrates a, form of the invention suitable for solvingpolynomials having real coefficients and complex roots, and

Fig. 3 illustrates the detailed connections of the potentometers whichare shown more or less diagrammatically in Figs. 1 and 2.

As hereinafter explained in connection with Fig. 3, the adjustmentsrequired to establish voltages representative of the various terms ofthe equation are elected byr means of potentiometers, the values of suchvoltages being determined by the adjustment of the potentiometers andthe polarities of the voltages being determined by the position of thepotentiometer switches. Allv ampliers herein except summing amplinersare adjusted to have unity gain.

The general polynomial is:

Case I.-Real coefficients and real roots. ao an all real; y=0; (nocomplex roots). Then, a root is any value of af:l such that:

The device of Fig. 1 isv shown as arranged to solve the above polynomialfor the case where n- ,-3. This. device includes a unit voltage sourcelil, a plurality of linear potentiometers l' to Il, three ampliers i8 to26, four summing resistors 2l to 24, a summing amplifier 25 and a nullindicator 26.

While the source I 0 is shown as a D. C. source, and A. C. source may beused with. satisfactory results. In order to simplify the drawing, thelinear potentiometers are shown somewhat diagrammatically. Their actualconnections are to be explained in connection with Fig. 3. The linearamplifiers t8 to 20, the summing amplier 25 and the null indicator 2tare like. those of the aforesaid patentsl (except that the ampliiiersinclude a feed-back resistor as shown in Fig. 3.) and are not describedherein in detail.

In operating the device.` of Fig. 1, the potentiometer I l is soadjusted as to apply through the resistor 2l to. the input of the.amplifier 25 a voltage which is representative. of the value of. theterm ao of the equation. Then the potentiometersv I.5,. Iiiy andr Il'are adjusted according to thev respectiveV valuesv of ai, a2. and aa.Under these conditions, simultaneous adjustment of the potentiometersi2, i3, lll which are ganged together as indicated by the. broken line2T, results in a null indication on the meter 26. Increasing the valueof :n from its` maximum negative value to its maximum positive valuewill produce as many such null indications as there are` distinct realroots. The values of these roots may be read from the indicatorassociated with the knob by which the values of a; are adjusted.

In the case involving real coefcients the complex roots,

a0 anY real; yeO- (complex roots) The polynominal is now:

ao-l-a1-(x+iy)+a2(+y)2l +an.(x-|-iy.)'=0

Expanding term by term We have:

Noting that (z`)2=1, ()3=-z', (174:1, etc. the coefcients of the variouspowers of :c and y may be tabulated as follows:

where Algebraic sums of the odd (real) columns and of the even(imaginary) columns are to be formed separately and a: and y are to beadjusted until both sums vanish. The total number of terms is obviously:

Fig. 2 shows the device as adapted to solve the equation in the casewhere ?1=3. This form of the device includes, in addition to thecomponents of Fig. l, linear potentiometers 28 to 39, linear ampliers 40to 46, a summing amplifier 41 and null indicator 48 for the imaginaryroots, and summing resistors 49 to 54.

All the a: potentiometers I2, I3, I4, 29, 3| and 35 are ganged togetheras indicated by the broken line 55. All the y potentiometers 28, 33 and31 are ganged together as indicated by the broken line 56. Above each ofthe linear potentiometers is an underlined legend, such as ao forexample, indicating the term in the expansion of the polynominal towhich the output voltage of the potentiometer is adjusted. Applied tothe various leads are legends indicating the terms which are representedby the voltages of the respective leads.

For example, a voltage representative of the value of y is applied tothe input of the amplifier 40 which has its output potentiometer 29adjusted according to the value 2.7:. As a result, the input voltage tothe amplier 4I is representative of the value of 21:11. Since the outputpotentiometer 30 of the amplifier 4I is adjusted according to the valueof a2, it follows that a voltage representative of the value of 2112 .ry is applied through the summing resistor I to the summing amplifier 41.

By following the same procedure in connection accesos* with the otherchannels which terminate at the summing resistors 2I to 24, 49, 50 and52 to 54, the eiects produced by these various channels is readilyunderstood without more detailed explanation.

In operating the device of Fig. 2 to determine the real and imaginaryroots of the polynomial, the procedure is as follows:

For real roots (l) Iadjust the control knob of the y potentiometers 28,33 and 31 to zero and (2) operate the control knob of the ganged a:potentiometers until the indicator 26 shows a null. As the value of :ris increased from its maximum negative value to its maximum positivevalue, there will be found as many nulls as there are distinct realroots, Athe values of such roots being indicated by the indicatorassociated With the o: control knob.

For complex roots (1) adjust the y control knob to some arbitrary valueof y other than zero, (2) adjust the .r control knob to produce a zeroreading on the indicator 26, and (3) preserving this zero reading on theindicator 26, adjust the :r and y control knobs in turn until a zeroreading appears on the indicator 48. At this point, the indicators.associated with the .r and y control knobs indicate the real andimaginary parts, respectively, of a complex root. By starting, in thismanner, from one end of the scale and proceeding uniformly toward theother end, there is obtained the roots (either real or complex) in theorder of their magnitude.

Fig. 3 shows in greater detail how the linear potentiometers are made toapply output voltages which are of such polarity and value as t0 trulyrepresent the various terms of the polynomial. In this iigure, theamplier I8 is taken as an example. Since the operation of all the otherampliers involves the same principle, such operation is readilyunderstood without detailed consideration.

The amplifier I8 has its anode and cathode connected respectively to thexed contacts 51 and 58 of a double throw switch 59. When the switch 59is in its lower closed position, the anode terminal 51 is connectedthrough a resistor 60 to the positive lead 6I and the cathode terminal58 is connected through the potentiometer I3 to the negative lead 62.Under these conditions, the left hand terminal of the potentiometer I3is more positive than its right hand terminal. When the switch 59 is inits upper closed position, the ano-de terminal 51 is connected throughthe potentiometer I3 to the positive lead 6I and the vcathode terminal58 is connected through the resistor 60 to the negative lead 62. Underthese conditi-ons, the left hand end of the potentiometer I3 is morenegative than its right hand end. Thus the closed positions of theswitch 59 determine the polarity of the output voltage applied from thepotentiometer I3 to the lead 63.

As indicated in Fig. 2, there is applied to the input of the amplier I 8from the potentiometer I2 a voltage which is representative of the valueof :11. Assuming the switch 59 (Fig. 3) to be in its upper closedposition, the voltage acre-ss the vpotentiometer I3 is of negativepolarity and has a Value proportional to the value of .'r. By adjustingthe movable contact 64 of the potentiometer I3 according to the value ofa: there is made available at the inputof amplier I9 a voltagerepresentative of the value of 3:2.

For producing a voltage representative of ma: at the input lead 65 ofthe summing amplifier 25 there is provided` means including a .switch 66which has its upper xed terminal-s Ii-I to 'Ill connected to the endterminals of the potentiometer I3 and the resis-tor 60. From thepotentiometer I3 there is applied to the potentiometer I5 a voltagewhich is representative of the Value of .r and has its polaritydetermined by the closed position of the switch 66 as explained inconnection with the switch 59. Assuming the movable contact 61 of thepotentiometer I5 t-o be adjusted accor-ding to the value of a1, there isproduced at the input 65 of the summing amplifier 25 a voltagerepresentative of the value of aix.

What the invention provides is an equation solver which functions tosolve polynomials for their real and complex roots by the directsynthesis method.

What is claimed is:

1. An equation solver for a polynomial equation of the type having realand imaginary terms and real and imaginary roots, said equation solverconsisting of a plurality of linear amplifiers, a plurality of linearpotentiometers, one .for each coeiiicient, one for each x, Iand one foreach y in said equation, all said coecient potentiometer sliders beingpositioned to represent cceilicients in said equation, al1 said :1:potentiometer sliders -being ganged together to be movablesimultaneously, all sai-d y potentiometer sliders being -ganged togetherto be movable simultaneously, means to apply a unit potential to a firsta: potentiometer, to a first y potentiometer and to a coefiicientpotentiometer representative of the coefficient ao, ya plurality offirst means to couple in cascade a number of said :c potentiometers andcoefiicient potentiom-eters to the slider of said first :c potentiometerto simulate the real terms am: to anx in said equation, a plurality ofsecond means to couple in cascade remaining ones of said :cpotentiometers, y potentiometers and coefficient potentiometers to theslider of said first y potentiometer to simulate the remaining real andimaginary terms of said equation, first and second summing means, rs-tand second null indicating means respectively connected to the outputsof said first and second summing means, all the sliders of thecoefficient potentiometers which are used to simulate said real termsbeing electrically -connected to said first summing mean-s, all thesliders of the coefficient potentiometers which are used to simulatesaid imaginary terms being electrically connected to said second summingmeans, and means to adj-ust said glanged potentiometer sliders and saidganged y potentiometer sliders whereby these sliders are positioned atvalues respectively representative of the real and imaginary roots ofsaid equation whenever both said null indicators indicate a null.

2. An equation solver as recited in claim 1 wherein said rst and secondplurality .of means to couple in cascade includes said linear feedbackamplifiers, the input to each of certain ones of said amplifiers beingconnected to the slider of an :c potentiometer, the output from each ofsaid certain ones of said amplifiers being connected to a following :cpotentiometer and a coefficient poten-tiometer, the input to each o fthe remaining ones of said amplifiers being connected to the sliders ofa y potentiometer, and the -output from each of said remaining ones ofsaid amplifiers being connected to a following :c potentiometer, ypotentiometer and a coefficient potentiometer.

ROBERT SERRELL. EDWIN A. GOLDBERG.

REFERENCES CTED rIhe following references are of record in the file ofthis patent:

UNITED STATES PATENTS Number Name Date 2,468,150 Wilcox Apr. 26, 19492,494,036 Darlington Jan. 10, 1950 2,519,223 Cheek Aug. 15, 19502,519,262 Lovell Aug. 15, 1950 OTHER REFERENCES An electromechanicalmethod for solving equations, Schooley, RCA Review, volume III, July1938, No. 1, pages 86-96.

An electrical algebraic equation solver, Herr and Graham, Rev. Sci.Inst., October 1938, volume 9, pages S10-315.

Analysis of problems in dynamics by electronic circuits, Ragazzini, IREProceeding, volume 35, No. 5, May 1947, pages 444-452.

Electrical -analogue computing, Mynall, Electrical Engineering, July1947, page 214.

Electronic Instruments, Greenwood et a-l., Radiation Laboratory Series,No. 21, McGraw- Hill Publishing Co., April 20, 1948, pages -122.

